Computational Fluid Dynamics for Designing Process Equipment

I NTERNATIONAL J OURNAL OF C HEMICAL R EACTOR E NGINEERING Volume 5

2007

Article A90

Three-Dimensional Gas-Liquid CFD Simulations in Cylindrical Bubble Columns Celso M. dos Santos∗

Renato Dionisio†

Henrique S. Cerqueira‡

Eduardo F. Sousa-Aguiar∗∗

Milton Mori††

Marcos A. d´Avila‡‡

∗

State University of Campinas, [email protected] State University of Campinas, [email protected] ‡ Petrobras, Cenpes, [email protected] ∗∗ Petrobras, Cenpes, [email protected] †† State University of Campinas, [email protected] ‡‡ State University of Campinas, [email protected] ISSN 1542-6580 †

Brought to you by | University of Saskatchewan Authenticated | 128.233.210.97 Download Date | 3/11/13 7:10 PM

Three-Dimensional Gas-Liquid CFD Simulations in Cylindrical Bubble Columns Celso M. dos Santos, Renato Dionisio, Henrique S. Cerqueira, Eduardo F. Sousa-Aguiar, Milton Mori, and Marcos A. d´Avila

Abstract Results from CFD simulations in a laboratory scale cylindrical bubble column with an internal diameter of 0.44 m under different operation conditions are presented. The effects of the continuous phase viscosity, bubble diameter and drag model were evaluated and the results were compared with experimental data found in the literature in three different gas inlet velocities. It was found that the approach used in this work provided physically consistent results, showing the transient effects in the column and good agreement with experimental data found in the literature for a homogeneous flow regime. Also, a case for a column with internals was simulated and a flow behavior qualitatively different from the column without internals was found. KEYWORDS: CFD, bubble columns, drag models, internals


dos Santos et al.: CFD Simulations in Bubble Columns

1. INTRODUCTION Bubble column reactors are found in many applications in chemical reaction engineering. They are encountered, among others, in biochemical processes (Kantarci, Borak, and Ulgen, 2005) and slurry reactors for Fischer-Tropsch synthesis (Maretto and Krishna, 1999; Ahón et al., 2005). One of the main characteristics that determine the proper performance of a bubble reactor is the fluid dynamics inside the column, which is very complex since it is a highly turbulent multiphase flow with complex chaotic dynamics (Deckwer, 1992; Cassanello et al., 2001). Both experimental and computational studies of bubble column flows are widely found in the literature and it is an active area of research. Different experimental techniques have been used in order to access the fluid flow properties inside bubble columns, such as particle imaging velocimetry (PIV) (Deen, Westerweel, and Delnoij, 2002; Delnoij et al., 1999; Chen and Fan,1992), radioactive particle tracking methods such as CARPT and RTP (Devanathan, Moslemian and Dudukovic, 1990; Xu et al., 2005), laser-doppler anemometry (LDA) (Pfleger and Becker, 2001; Kulkarni, Ekambara and Joshi, 2007), ultrasound (Zheng and Zhang, 2004), high speed video (Zaruba et al., 2005), optical probes (Chaumat, Billet and Delmas, 2006), among others. Advances in numerical studies of bubble column flows have seen an increasing growth mainly due both to the larger availability of low cost and high speed computers and to the advancement of numerical techniques, which is capable to perform three-dimensional simulations of multiphase flows in complex geometries (Joshi and Ranade, 2003). Despite the recent advances in both experimental and numerical techniques, flow in bubble columns is far from being fully understood (Jakobsen, Lindborg and Dorao, 2005). However, the proper use of the available tools can be very useful in developing scale-up strategies and understanding their flow behavior (Krishna, 2000). A large number of gas-liquid simulation works were performed by approaching the flow as being twodimensional or axisymmetric three-dimensional and, despite the limitations of these approaches, recent articles have shown their feasibility and usefulness for both rectangular (Bech, 2005) and cylindrical columns (Joshi, 2001; Ekambara, Dhotre and Joshi, 2005). These methods, although limited, are able to provide satisfactory predictions of time-averaged properties and were used to model bubble column reactors flow in laboratory (Sanyal et al., 1999) and industrial scales (Krishna, Van Baten and Urseanu, 2000). However, there are limitations on such approaches and they are often inadequate to describe the transient flow, which is an important aspect of the fluid dynamics inside bubble columns (Krishna and Van Baten, 2004). Starting in the last decade, a large number of fully threedimensional gas-liquid flow simulations in bubble columns have been published and are reviewed in several articles (Jakobsen et al., 1997; Sokolichin, Eigenberger and Lapin, 2004; Jakobsen, Lindborg and Dorao, 2005; Rafique, Chen and Dudukovic, 2004). Nowadays, it is widely accepted that such simulations provide the best account on the fluid dynamical behavior in bubble columns (Sokolichin and Eigenberger, 1999). Most of these studies were performed using the Eulerian-Eulerian approach, but contributions can also be found in the literature using Eulerian-Lagrangian approach (Lain, Broder and Sommerfeld, 1999; Sommerfeld, Bourloutski and Broder, 2003; Zhang and Ahmadi, 2005). Computational fluid dynamics (CFD) studies have been carried out aiming the scale-up of slurry reactors (Van Baten, Ellenberger and Krishna, 2003), the modeling of the churn-turbulent flow regime, the presence of catalyst through three-phase flow simulations (Matonis, Gidaspow and Bahary, 2002), the break-up and coalescence of bubbles (Olmos et al., 2001; Chen, Dudukovic and Sanyal, 2005; van den Hengel, Deen and Kuipers, 2005; Sha, Laari and Turunen, 2006), the mass transfer (Wiemann and Mewes, 2005) and chemical reactions (Rigopoulos and Jones, 2003; Van Baten and Krishna, 2004), among others. However, there are important issues that have been investigated and a consensus has not been achieved. One can highlight the turbulence and drag modeling. Studies in geometry can be found in evaluation of columns of different sizes in order to test and obtain scale-up correlations based on CFD simulations (Krishna et al., 2001; Van Baten and Krishna, 2004) and the effect of gas sparger (Ranade and Tayalia, 2001; Akhtar, Tadé and Pareek, 2006; Dothre and Joshi, 2007) and internals (Larachi et al., 2006) in bubble column flows. Lattice Boltzmann simulations (Theodoropoulos et al., 2004) has been applied to study flow of bubbles, but it still have limited applicability in systems of industrial interest. In the present article, we present fully three-dimensional, gas-liquid simulations in a cylindrical bubble column of laboratory scale using the Eulerian-Eulerian approach. The aim of this article was to present the effects of important parameters in CFD simulations such as bubble size, continuous phase viscosity and drag models. Simulations were performed using the commercial CFD package CFX 5.7 from ANSYS. The geometry was chosen in order to compare simulation results with experimental data published by Chen et al. (Chen et al., 1998; Chen et


1

International Journal of Chemical Reactor Engineering

2

Vol. 5 [2007], Article A90

al., 1999) which consisted in a cylindrical column with 0.44 m of inner diameter with liquid initially filled up to 1.70 m in height. The effects of drag models, bubble diameter and continuous phase properties in three different gas inlet velocities were evaluated. The drag models used were the Ishii-Zuber (Ishii and Zuber, 1979) and Grace (Clift, Grace and Weber, 1978), which consider bubble deformation effects. The Schiller-Naumann drag model for rigid spheres was also evaluated. Turbulence was considered only for the continuous phase, where the standard k-epsilon model was used. The lift, Magnus and added mass forces were neglected. Bubble breakup and coalescence were not considered. The fluids modeled were water and viscous oil and the gas phase was air at room temperature. The bubble diameter was estimated using a literature correlation that estimates the bubble diameter in perforated plates based on the inlet characteristics, such as inlet flow rate and diameter of holes in the perforated plate. Numerical results of time-averaged properties were compared with experimental data. Results have shown that the approach used in this work provided was able to describe the transient flow behavior inside the column. Good agreement of time-averaged gas holdup and axial liquid velocity with experimental data was obtained for the cases where the churn-turbulent flow was not achieved. Both Grace and Ishii-Zuber models provided good predictions when the flow was in the homogeneous regime. It was observed that the continuous phase properties, i.e., viscosity and interfacial tension, strongly affect the flow inside the bubble column due. It was also found that in the simulations, there is a considerable effect of the bubble size, which points out the importance of using appropriate correlations to estimate bubble size for each flow condition, which sometimes is not performed in CFD studies in bubble columns. Also, a case for a column with internals was simulated. The presence of internals inside the column represents the cooling tubes that are inserted in bubble column reactors. Results for the geometry with inserted tubes showed that the modeling presented in this article can provide physically consistent results. It was found a flow behavior qualitatively different when compared to the column without internals.

2. MATHEMATICAL MODELING The flow in the bubble column was modeled using the Eulerian-Eulerian approach, which considers the dispersed phase as an inter-penetrating continuum. Thus, the continuity and momentum equations are solved for each phase and is given, respectively by

∂ (ε k ρ k ) + ∇• (ε k ρ k u k ) = 0 ∂t and ∂ (ε k ρ k u k ) + ∇ • (ε k (ρ k u k u k )) = ε k ∇Pk + ∂t

(

(

+ ∇ • ε k μ k ∇u k + (∇u k )

T

))+ M

α

(1)

(2)

+ ρk g

Here, ρ is the density, ε is the volume fraction, u is the velocity vector, μ is the viscosity and the subscript k indicates the phase. Note that in the above equations, the inter-phase mass transfer is not considered. The term Mα corresponds to the inter-phase forces, which is the sum of all forces given by D L LUB VM M α = ∑ M αβ = M αβ + M αβ + M αβ + M αβ + M TD αβ + ... ,

(3)

β≠ α

D

L

where α and β indicates the phases (continuous and dispersed). In the above equation, M αβ is the drag force, M αβ LUB

is the lift force, M αβ

VM

TD

is the wall lubrication force, M αβ is the added mass force and M αβ is the turbulence

dispersion force. In the present article we considered that the only inter-phase momentum transfer acting in the



3

system was the drag force. Although the other interactions may be affecting the flow behavior, it is well-known that the drag interactions is the main inter-phase force affecting flow in bubble columns for a given flow condition and geometry (Jakobsen et al., 1997). Turbulence was considered for the continuous phase and a standard k-epsilon turbulence model was used and a discussion on the applicability of this model can be found in the article of Sokolichin and Eigenberger, (1999). This model is based in the energy generation κ and dissipation epsilon due to the turbulence. This model was chosen since it has shown to provide satisfactory results in bubble columns CFD simulations (Krishna and Van Baten, 2004; Sokolichin and Eigenberger, 1999). The standard parameters values of C μ = 0.09 ,

C ε1 = 1.44 , C ε 2 = 1.92 , σ k = 1 and σ ε = 1.3 were used. The drag force term is expressed, for the continuous phase mathematically by D M αβ =

CD A gl ρ l (u l − u g )u g − u l 8

(4)

In the above equation the subscript c indicates the continuous phase, CD is the drag coefficient and A gl = 6ε g / d g is the interfacial area per unit of volume of an ensemble of particles of diameter dg and volume fraction εg. The expression for the drag coefficient CD varies according to the multiphase system. For the case of an isolated spherical particle at low Reynolds number, the drag coefficient can be obtained analitically from the NavierStokes equations, where CD = 24/Rep. For bubbly flows, the expression for CD must take in to account bubble deformation and internal flow and there is no simple analytical expression for this case. Therefore, empirical and semi-empirical correlations are used to calculate CD. In this article, the models from Ishii and Zuber, (1979) and Grace (Clift, Grace and Weber, 1978) were used. Both models consider different expressions and/or values for the drag depending on the bubble shape, which can be spherical, ellipsoidal, or a spherical cap. Also, these models are able to take into account the effects of gas volume fraction in the system. Each bubble shape has a particular expression for CD and it is obtained by the following conditional expressions: ellipse C dist , C cap D = min(C D D )

(5a)

and

C D = max(C sphere , C dist D D ).

(5b)

cap C sphere , C ellipse and C D are the drag coefficients for spherical, ellipsoidal and cap bubbles. For D D sphere both Grace and Ishii-Zuber (IZ) models in the range of particle Reynolds number for the system studied, C D are

Here,

respectively given by ,Grace C sphere = D

24 Re p

and , IZ C sphere = D

(6)

(

24 1 + 0.15 Re 0m, 687 Re

)

Here, Re m = ρ l u g − u l / μ m , where μ m = μ l (1 − ε g / ε dm )

(7) −2 , 5 ε gμ∗

is a bulk viscosity and ε dm is the

maximum packing of the dispersed phase. Equation (7) corresponds to the Schiller-Naumann (SN) model for drag coefficient of rigid particles. cap

The values of C D for Grace and IZ models are



4

,Grace C cap = C D∞ = D

8 3


(8)

and

, IZ C cap = (1 − ε g ) C D∞ . D 2

(9) ellipse, IZ

In the case of ellipsoidal particles, the drag coefficient for the IZ model C D , IZ C ellipse = E (rd )C dilute D D

is given by

(10) dilute

In the above equation, C D is the drag coefficient for an ellipsoidal bubble in a dilute system and E(r d) is a function that takes into account the deformation effects and is respectively given by

C dilute = D

2 12 Eo 3

( 11 )

(1 + 17.67f (r ) ) E (ε ) = 6

g

g

7

18.67f (ε g )

( 12 )

,

μl (1 − ε g )12 and the dimensionless number Eo = (g Δρ d g2 ) / σ is known as the μm Eotvos number, where Δρ = ρ l − ρ g .

( )

where f ε g =

ellipse,Grace

For the Grace drag model, the equation for C D ,Grace C ellipse = D

is given by

4 gd g Δρ 3 U T 2ρ l

(13)

where UT is the terminal rise velocity of a single bubble and it is given by

UT =

μl M −0.149 (J − 0.857 ) ρl d g

(14)

In the above equation, M is a dimensionless parameter known as the Morton number, which takes into account buoyancy and interfacial effects in the bubbly flow and J is an empirical parameter. These are respectively given by

M=

( 15 )

μ l4 gΔp , ρ 2σ3

and

⎧94 H 0.751 ⎪ J=⎨ ⎪3.42 H 0, 441 ⎩

2 < H ≤ 59.3 H > 59.3


( 16 )


4 −0.149 ⎛ μ l ⎜⎜ In the above equation H = E 0 M 3 ⎝ μ ref

5

⎞ ⎟⎟ ⎠

−0.14

, where μref= 0.0009 kg/(m.s) is a reference viscosity.

In order to consider the effects of high gas volume fractions, the value of the drag coefficient for the Grace model can be adjusted by the following expression: p

,dense C Grace = ε l C Grace D D

(17)

Here, the exponent p is a parameter that is chosen depending on the system. For systems of small bubbles, p is on the range of -1 to -0.5, since small bubbles tend to rise more slowly at high volume fractions, due to an increase in the effective mixture viscosity. In the case of large bubbles p is on the range of 1 to 2, since they tend to rise faster at high gas volume fractions, because they are dragged along by the wakes of other bubbles (CFX User guide, 2005).

3. SIMULATION SET UP 3.1 Geometry and fluid properties The geometry of the bubble column used in this work consisted in a cylindrical vertical column with inner diameter d=0.44 m and total height H = 2.43 m, with the column initially filled up to a height of ho=1.7 m. This geometry corresponds to the column used in the experimental studies of Chen et al., (1998) and Chen et al., (1999) where time-averaged velocity and gas volume fraction fields in a perforated plate bubbling system were measured, using CARPT-CT (Computer Automated Radioactive Particle Tracking – Computer Tomography). A scheme of this geometry is shown in Figure 1. The gas inlet is on the bottom of the column, which enters with a superficial velocity Us = Q/A, where Q is the volumetric gas flow rate in the entrance and A is the cross section area of the column. The top of the column is opened and the flow is isothermal. Simulations were performed considering the gas phase as air and two different continuous phases. The low viscosity one has physical properties of water, whereas the high viscosity liquid has physical properties of Drakeoil® 10 (Van Waters & Rogers Inc.). The fluid properties are shown in Table 1. Three different inlet velocities Us were simulated: 0.02, 0.05 and 0.1 m/s. In water/air system Us=0.02 m/s correspond to the homogeneous flow regime, whereas Us=0.05 m/s and Us=0.1 m/s correspond, respectively, to the transition and churn-turbulent flow regimes. In the oil/air system, only Us=0.02 m/s can be considered to be in the homogeneous/transition system (Chen et al., 1999). Above this value, the system is in the churn-turbulent regime.



6


Table 1. Fluid properties. Water Specific mass (kg/m) Viscosity (kg/(m.s)

H

Air

997 8.9×10

Oil

1.185 -4

1.831×10

860 -5

3,0×10-2

Interfacial tension (N/m) 0.072

h

h0

0.035

One important parameter in bubble column modeling is the bubble size. This is mainly dependent on the inlet flow rate Q, the sparger characteristics, the continuous phase physical properties, such as the viscosity and the specific mass. Measurements of bubble size and bubble size distributions in systems with high gas holdup such as the one studied here is a difficult task and, to our knowledge, precise experimental data of bubble sizes were not found in the literature.

Us Figure 1. Geometry used in the simulations based on the experimental set up of Chen et al., (1998) , where H=2.43 m, h0= 1.70 m and h is the height in the column.

In CFD studies of bubble columns, it is usually assumed a single arbitrary value of the bubble, which may correspond to estimated values based on observations and on the column properties (Sanyal et al., 1999), or it is avoided in the modeling by finding a suitable empirical equation for the drag based on the column characteristics (Krishna and Van Baten, 2004).

In the present paper, the bubble size were estimated using an empirical expression for air bubbles produced in perforated plates given by Treybal, (1980).

d g = 0,0071 Re o

−0 , 05

(18)

where dg is the bubble diameter and Re 0 is the air Reynolds number in the orifice. Therefore, the influence of the continuous phase (water or oil) is not considered in the calculations, which might be a major simplification. However, Equation (18) was found to be the most suitable correlation and comparisons with the values obtained by this correlation and estimates from other CFD articles published were in good agreement (Sanyal et al., 1999). Values of bubble diameter found for different inlet superficial velocities U s using Equation (18) can be seen in Table 2. Table 2. Bubble diameter at different inlet superficial velocities. dp x 103 (m)

U (m/s) 0.02 0.05 0.1

6.7 4.1 4.0

3.2 Mesh In order to perform the simulations, a mesh composed of hexahedral elements was built using the software ICEM CFD. Refinement tests were performed and the one that was used consisted of 120 elements in the height, 27 elements in the radial direction and 24 elements in the azimutal position, with a total of approximately 61000 elements.



7

The element dimensions are similar or smaller when compared with other three-dimensional CFD studies found in the literature for columns with similar dimensions. Figure 2 shows a view of the mesh. Details on the mesh evaluation procedure can be found elsewhere (Santos, 2005) .

(a)

(b)

The dimensions of the column with internals are the same as the one presented in Figure 1, with 16 cylindrical tubes of 2.54x10-2 m diameter equally spaced inside the column. The mesh for this geometry was built using the package ICEM CFD and it is shown in Figure 3. Due to its complexity, the total number of elements was approximately 600000, which significantly increased the simulation time.

Figure 2. Three-dimensional hexahedrical mesh used in this work (without internals): (a) cross-section and (b) longitudinal view.

(a)

(b)

Figure 3. Three-dimensional mesh of the column with internals used in this work: (a) cross-section and (b) longitudinal view (note that it is not shown in the whole extension in order to make visible the hexahedrical elements). The highlighted area shows the mesh refinement near the tube walls.



8


3.3 Simulation parameters, initial and boundary conditions Simulations were performed considering that the fluid in the column was initially quiescent at a height ho = 1.70 m. Both phases were considered to obey the non-slip condition at the walls. The gas inlet was considered to be uniform with constant Us. It was considered that the gas inlet occurred within the 56% of the total area of the column crosssection. This procedure was used by Van Baten and Krishna (2004), where better results were achieved for perforated plates in cylindrical columns. In the column top, an opened boundary with atmospheric pressure was considered. In order to guarantee convergence, the gas inlet was slowly changed in steps of 0.001 m/s after every 1 second until the inlet velocity reached the column operation inlet superficial velocity. In the simulations, the time steps used were 2 iterations of 0.005 s and 990 iterations of 0.001 s, followed by iterations of 0.01 s until the simulation completion. The interpolation in time used was a first order Euler scheme. A high order upwind scheme was used for the convective term interpolation. The data was recorded for timeaveraged analysis for times higher than 20 seconds after the operation flow rate was reached. The convergence criteria used was that the root mean-square deviation for all properties should be less than 5x10-5. Table 3 shows the cases studied in this work. Simulations were performed using parallel processing with two Pentium 4 processors. Simulation times for the cases without internals ranged from 360 to 720 hours of simulation. The case with internals took approximately three months of simulation. Table 3. Simulation cases. In the case code, the first letter is the initial of the continuousphase, the second number is the inlet velocity, the third indicates the bubble diameter and the last characters indicate the drag model. Case

Us (m/s)

O26G W26G O24G W54G O54G W56G W54I W54G5 W54G1 W54SN W104G O104G W54GI

0.02 0.02 0.02 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.1 0.1 0.05

Continuous Phase Oil Water Oil Water Oil Water Water Water Water Water Water Oil Water

dg x103 (m)

Drag Model

Geometry

6.75 6.75 4.17 4.17 4.17 6.75 4.17 4.17 4.17 4.17 4.00 4.00 4.13

Grace [Eq.] Grace [Eq.] Grace [Eq.] Grace [Eq.] Grace [Eq.] Grace Ishii-Zuber [Eq.] Grace, p = -0.5 Grace, p = 1 Schiller-Naumann Grace Grace Grace

W/o internals W/o internals W/o internals W/o internals W/o internals W/o internals W/o internals W/o internals W/o internals W/o internals W/o internals W/o internals With internals

4. RESULTS AND DISCUSSION 4.1. Transient results As pointed out in section 1, the fluid dynamics in bubble columns is highly transient. Therefore, there is a strong dependence on the system flow properties in space and time. Since most of the experimental studies found in the literature present time-averaged data, it is important to evaluate the transient behavior of the column to make appropriate time-averaging procedures in order to compare numerical data with experimental results. Figure 4 shows maps of liquid axial velocity Vy and gas volume fraction ε g at different times from 30 to 80 seconds for the case W54G (see Table 3). It can be seen that the simulation captures the transient behavior of the column, where there is a similarity between the velocity and hold-up profiles. This means that the location of high gas volume fraction correspond to the location of the high liquid velocities. This is physically consistent and expected since the fluid dynamics in the column occurs due to the momentum transfer from the gas to the liquid. Also, it can be seen, by following the traced line, that the initial level of liquid increased due to the injection of gas, and it remained constant after 8 seconds. The similarity in the ε g and vy profiles is also found in the literature, where in the center of the column ε g and vy assume higher values due to the drag effects (Sanyal et al., 1999; Chen et al., 1998).



9

εφgd

1.0

0.75 0.5 0.25 0.0

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

vy (m/s) 1.0

0.5 0.0 -0.5 -1.0

Figure 4. Maps of gas volume fraction (a-f) and continuous phase axial velocity (g-l) for water-air at U s = 0.05 m/s at different times: a = 30 s, b = 40 s, c = 50 s, d = 60 s, e =70 s, f=80 s; g=30 s, h=40 s, i=50 s, j=60 s, l=70 s, m=80 s. Figure 5 presents profiles of axial liquid velocity vy, gas volume fraction εg, time-averaged axial liquid velocity vy,avg and time-averaged gas volume fraction εg,avg for the case W54G in the center of the column cross section at a height h = 1.32 m. The oscillatory behavior of the flow is clearly observed, where the maximum and minimum of vy and εg are practically coincident. This oscillatory behavior is chaotic and a characteristic oscillation frequency was not found. Also, the oscillations for v y seem to be smoother than for εg. The time-averaged data vy,avg and εg,avg do not oscillate after a period of time, which can be seen in the figure after approximately 80 seconds. After this time, the values appear not to vary considerably.




Axial Velocity(m/s) Axial Velocity

1,2

0,4 0,3

1

0,2

0,8

0,1

0,6

0

0,4

-0,2

-0,1 -0,3

0,2

-0,4

0 -0,2

0

20

40

60

Average Axial Velocity Average Gas Holdup time.

Gás Holdup Gas Holdup

10

Time

80

100

120

-0,5 -0,6

Axial velocity Gas Holdup

Figure 5. Values of vy , εg , vy, avg and εg,avg for the case W45G in the point (0; 1.32 m; 0) as a function of

This oscillatory behavior is also observed for different inlet velocities as can be seen in Figure 6, where gas volume fraction profiles were obtained for different cases. Figures 6a, c and e correspond to the water-air system and Figures 6 b, d and f are for oil-air system. It can be seen that when Us increases, the oscillation frequency increases and the time where the average value does not vary significantly decreases with the increase of Us. This observation can be used to set the appropriate simulation time in order to compare time-averaged results with experimental data. Unfortunately, experimental data of the oscillatory behavior was not available in order to compare with the simulations.



1

1

(a)

0.8

Gas Holdup

Gas Holdup

11

0.6 0.4

0.4

0

0 0

20

40

60

80

100

0

120

Time (s)

1

20

40

1

(c)

0.8

Gas Holdup

Gas Holdup

0.6

0.2

0.2

0.6 0.4 0.2

60

80

Time (s)

100

120

80

100

120

80

100

120

(d)

0.8 0.6 0.4 0.2

0

0 0

20

40

60

Time (s)

80

100

120

0

(e)

1

20

40

60

Time (s)

(f)

1

0.8

Gas Holdup

Gas Holdup

(b)

0.8

0.6 0.4 0.2

0.8 0.6 0.4 0.2

0

0 0

20

40

60

Time (s)

80

100

120

0

20

40

60

Time (s)

Figure 6. Values of εg and εg,avg in the point (0, 1.32 m, 0) as a function of time for the cases (a) W104G, (b) O104G, (c) W54G, (d) O54G, (e) W26G and (f)O26G.

4.2. Time-averaged results Figure 7 shows maps of gas volume fraction for both systems, when Us = 0.05 m/s. It can be seen that the timeaveraged hold-up presents, for almost the whole extent of the column, a profile that reaches a maximum by the center of the column and decreases as it approaches the wall. However, it can be also noted that the holdup profile is not axysymmetric. These observations agree qualitatively with the experimental data of Chen et al., (1998) and from other experimental works (Deen, Westerweel, and Delnoij, 2002; Devanathan, Moslemian and Dudukovic, 1990). This also shows that in order to properly describe the fluid dynamics in bubble columns using CFD it is necessary to perform fully three-dimensional simulations. Even the so-called axisymmetric 3-D simulations wouldn't be able to describe properly the flow patterns. However, simplified modeling has been successfully used to predict averaged properties and could be used whenever such information is desired (Ekambara, Dhotre and Joshi, 2005). From the time-averaged axial liquid velocity maps it can be observed that the time-averaged flow pattern consists on ascendant flow in the central region of the column and descendent in the region closer to the walls. This same pattern was also observed for the other cases studied and qualitatively agrees with the experimental observations found in the literature (Sanyal et al., 1999).



12

εg,avgφd


vy,avg vy (m/s) (m/s)

1.0

1.0

0.75

0.5

0.5

0.0

0.25

-0.5

0.0

-1.0

(a)

(b)

(c)

(d)

Figure 7. Time averaged gas volume fraction and axial liquid velocity maps at h = 0.89 and Us = 0.05 m/s for (a and c) water-air, and (b and d) oil-air systems. 4.2.1 Bubble size and continuous phase effects As seen previously, the time-averaged values of axial liquid velocity and gas volume fraction reach an average constant value after a determined simulation time. The profiles presented here correspond to the azimutal average of a given time-averaged property. Figure 8 shows results of gas holdup for water-air system with inlet superficial velocities of 0.02, 0.05 and 0.1 m/s. It can be seen a good agreement between simulation and experimental data from Chen et al., (1998) and Chen et al., (1999) for Us =0.02 and 0.05 m/s. For the case of Us=0.1 m/s, the simulated profiles seems to be independent on the droplet size, and a large discrepancy between simulation and experimental results can be observed. The reason for this discrepancy is the transition from homogeneous to heterogeneous (or churn-turbulent) regime. For water-air system, the transition to churn-turbulent regime occurs when Us = 0.0045 m/s (Krishna, 2000). This regime is characterized by bubble break up and coalescence leading to a polydispersed population of bubbles. Since our simulation model does not contemplate these effects, it was expected that the simulation results wouldn't provide good agreement with experimental data for this case. Therefore, the CFD model used in this article provided good results for homogeneous and transition regimes and is not appropriate for churnturbulent regime. Some recent articles in the literature have handled the churn-turbulent regime by using suitable empirical drag models (Krishna and Van Baten, 2004) and by modeling the break up and coalescence process (Chen, Dudukovic and Sanyal, 2005). CFD simulations using break up and coalescence models are currently in progress by our group.



13

Chen et al. (1998); 0.02m/s Chen et al. (1998); 0.05m/s Chen et al. (1998); 0.1m/s simulation; 0.02m/s simulation; 0.05m/s simulation; 0.1m/s

0.4

Axial Liquid Velocity (m/s)

0.5

0.4

0.3 0.2 0.1 0

0.2 0.1 0 -0.1 -0.2 -0.3

0

0.2

0.4

0.6

0.8

Simulation; 0.05m/s Simulation; 0.02m/s Chen et al.; 0.02m/s

0.3

1

0

0.2

0.4

0.6

0.8

1

r/R

r/R

Figure 8. Gas volume fraction profiles for h = 0.89 m for Figure 9. Axial liquid velocity profiles for h = 1.32 m air-water system. Filled circles correspond to for air-water system. Filled circles correspond to experimental data by Chen et al. experimental data by Chen et al. Figure 9 shows comparisons of simulations with experimental data of time-averaged axial velocity for the same system shown in Figure 8. It can be seen that there is a good agreement with experimental data for Us = 0.02 and 0.05 m/s and both profiles exhibit the behavior of a velocity maximum in the center of the column and a descendent flow in the region close to the walls. Holdup profiles for the oil-air system are shown in Figure 10. It is well known that the inlet superficial velocity where the transition for the churn-turbulent regime occurs decreases when the continuous phase viscosity increases. Therefore, the case of Us =0.1 m/s was not simulated. It can be seen a good agreement between experimental and simulation data for Us =0.02 m/s. However, for Us =0.05 m/s there is a substantial overprediction and the bubble size affects considerably the gas volume fraction profile. This discrepancy with experimental result for this case is explained by the fact that the flow might be already in the churn-turbulent regime. For Us = 0.02 m/s, the best prediction is found when dg = 6.75x10-3 m, which is the value found using Equation (18). Velocity profiles for Us = 0.02 m/s and 0.05 m/s is shown in Figure 11.

0.5

0.5

Chen et al. (1998); 0.02 m/s Chen et al. (1998); 0.05 m/s simulation; 0.02 m/s simulation; 0.05 m/s

Axial Liquid Velocity (m/s)

0.4 0.3 0.2 0.1 0

0

0.2

0.4

0.6

0.8

Simulation; 0.05 m/s Simulation; 0.02 m/s Chen et al.; 0.02 m/s

0.4

1

r/R

Figure 10. Gas volume fraction profiles for h = 1.32 m for air-oil system for (a) Us = 0.02 m/s, and (b) Us = 0.05 m/s. Filled circles correspond to experimental data by Chen et al., (1998)

0.3 0.2 0.1 0 -0.1 -0.2 -0.3

0

0.2

0.4

0.6

0.8

1

r/R

Figure 11. Axial velocity profiles for h = 1.32 m for airoil system for Us = 0.02 m/s, and Us= 0.05 m/s. Filled circles correspond to experimental data by Chen et al.



14


4.2.2 Drag model influence The dependence of the averaged properties in the bubble size is directly related to the drag model used. As presented in section 2, the drag model requires a known bubble size. In some articles found in the literature, the bubble size is estimated to a single value independent of the inlet velocity and continuous phase (Sanyal et al., 1999;Krishna, Van Baten and Urseanu, 2000). Recent approaches have used empirical correlations, such as the one used in this work, relating the entrance geometrical characteristics, the fluids properties and the inlet flow rate (Chen, Dudukovic and Sanyal, 2005). Figure 12 shows a comparison using different drag models and experimental data for gas holdup profiles for water/air system at Us = 0.05 m/s. It can be observed that there is an influence of the drag model in the simulation results. The IZ model overpredicts the experimental value, whereas the Grace model underpredicts the data. As seen before, there is also the effect of the bubble size, which was set according to Equation (18). 0.3

Chen et al. (1998) Grace; p = 0 Grace; p = 1 Grace; p = -0.5 Ishii-Zuber Schiller-Naumann

0.25 0.2 0.15 0.1 0.05 0

0

0.2

0.4

0.6

0.8

1

r/R

Figure 12. Gas volume fraction profiles for h = 0.89 for air-oil system and Us = 0.05 m/s. Filled circles correspond to experimental data by Chen et al., 1998. As exposed in section 2, the Grace model (Eq. 17) can be adjusted to a suitable exponent p, which considers the gas concentration effects. Figure 12 also shows the data using three different values of p: 0, 1 and -0.5. It can be seen that the best agreement with experimental data was found for p = -0.5, which is in excellent agreement with experimental data. A negative exponent p leads to an increase in the drag coefficient CD, as seen in (Eq. 17). This has lead to an increase in the gas holdup prediction, which improved the agreement to the experimental value, for this case. However, the appropriate value of p can vary depending on the system characteristics.

4.3. Column with internals Figure 13 shows instantaneous maps of gas volume fraction and axial liquid velocity. It can be seen the complex flow behavior due to the presence of the internals. Gas holdup for the column with internals together with the counterpart without internals is shown in figure 14a. It can be seen that the profile shows a different dependence in the radial position, showing an effect of the presence of internals in the qualitative behavior of the gas holdup. However, it can be seen that the holdup values are in the same range as the column without internals. In figure 14b it is shown a cross-section volume fraction map at h=0.89 m. It can be seen that there is high gas concentration nearby the internals. This shows that, in CFD simulations, the presence of internals should be considered when it is desirable to obtain detailed information regarding the fluid dynamics of the column.



15

φd

1.0

0.75 0.5 0.25 0.0

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

vy (m/s) 1.0

0.5 0.0 -0.5 -1.0

Figure 13. Maps of gas volume fraction (a-f) and continuous phase axial velocity (g-l) for water-air at Us=0.05 m/s at different times for the column with internals: a = 30 s, b = 40 s, c = 50 s, d = 60 s, e =70 s, f=80 s; g=30 s, h=40 s, i=50 s, j=60 s, l=70 s, m=77 s. Gas Volume Fraction

0.3

0.15

Chen et al. (1998) without internals with internals

0.25

0.125

0.2 0.15

0.075

0.1

0.037 0.0

0.05 0

0

0.2

0.4

0.6

0.8

1

r/R

(b)

(a)

Figure 14. Gas volume fraction (a) profile and (b) cross-section map for the case W54GI, showing the influence of the presence of internals.



16


5. CONCLUSIONS This article presented a gas-liquid flow CFD study of a laboratory scale bubble column, where the effects of the continuous phase, the bubble size and the drag model were evaluated. Also, a simulation case with a column with internals was obtained. Results have shown that the modeling used is appropriate for the homogeneous flow regime in both low liquid viscosity (water-air) and high liquid viscosity (oil-air) systems. It was found that the simulation approach used in this work was able to capture the transient fluid dynamics. Time averaged results have shown that the flow field obtained is not axysymmetric, which was in qualitative agreement with experimental observation reported in the literature. It was found that the assigned bubble size in the simulations considerably affects the simulation results. An analysis based on the values of CD has shown that this effect can be correlated with a characteristic value of CD. Therefore, it can be concluded that in the absence of experimental data, a suitable bubble size should be used based on estimates using correlations found in the literature. Both Grace and Ishii-Zuber drag models were tested and it was found to provide satisfactory agreement with experimental data. It was found that the value of the exponent p = -0.5 provided excellent agreement with the experimental data for water-air system. Simulation for the column with internals provided numerical results regarding the flow field, which was found to be qualitatively different from the column without internals. However, holdup profiles showed that the holdup values obtained are in the same range when compared to its counterpart without internals.

ACKNOWLEDGEMENTS The authors gratefully acknowledge Fundação de Amparo à Pesquisa do Estado de São Paulo - Fapesp (03/01892 and 03/11208-0) for the financial support.

SIMBOLS USED Alg A CD CD ∞

interfacial area between gas and liquid phases, m2 column cross section area, m2 drag coefficient, kg.m3/s drag coefficient of dilute system, kg.m3/s

CDcap C Dellipse C Dshphere C DDilute

drag coefficient for spherical cap, kg.m3/s

cμ cε1 cε2 d dg Eo

g

h H h0 M

Mα M αβ D M αβ

drag coefficient for ellipsoidal particle, kg.m3/s drag coefficient for spherical particle, kg.m3/s drag coefficient for dilute sistem, kg.m3/s turbulence model constant turbulence model constant turbulence model constant column diameter, m bubble diameter, mm Eotvos number gravitational acceleration, m/s2 bubble column position, m bubble column height, m initial liquid height, m Morton number inter-phase forces in phase α, kg.m/s interfacial forces between phases α and β, kg.m/s drag force, kg.m/s



L M αβ

lift force, kg.m/s

LUB

M αβ

wall lubrification force, kg.m/s

VM

M αβ

TD M αβ

p Pk

Q

Rep Rem uk Us vy vy,avg UT Greek letters εg εg, avg εdm εk μc μeff μk

μm μ ref

virtual mass force, kg.m/s turbulence dispersion force, kg.m/s volume fraction exponent in Grace drag model turbulence production, kg/(m.s) flow rate, kg/s particle Reynolds number modified Reynolds number velocity of phase k, m/s inlet superficial gas velocity, m/s axial liquid phase velocity, m/s axial velocity average of liquid phase, m/s terminal rise velocity of gas phase, m/s

gas volume fraction time-averaged gas volume fraction dispersed phase maximum packing volume fraction of phase k continuous phase viscosity, kg/(m.s) effective viscosity, kg/(m.s) viscosity of phase k, kg/(m.s) modified viscosity , kg/(m.s) reference viscosity, kg/(m.s)

ρc ρk σk, σ1, σε σ

density of continuous phase, kg/m3 density of phase k, kg/m3 constants superficial tension

Subscripts g l k α, β

gas phase liquid phase anything phase phase

REFERENCES Ahón, V. R., Costa Jr., E. F., Monteagudo, J. E. P. , Fontes, C. E., Biscaia Jr., E. C., Lage, P L.C. “A comprehensive mathematical model for the Fischer–Tropsch synthesis in well-mixed slurry reactors”, Chem. Eng. Sci., Vol. 60, 677–694 (2005). Akhtar, M.A., Tadé, M. O., Pareek, V. K. “Two-Fluid Eulerian Simulation of Bubble Column Reactors with Distributors”, J. Chem. Eng. Japan, Vol. 39, 831-841 (2006) . ANSYS, Inc., “CFX-5 Users Guide”, Southpointe, 275 Technology Drive, Canonsburg, PA 15317, USA, (2005).


17

18



Bech, K. “Dynamic simulation of a 2D bubble column”, Chem. Eng. Sci., Vol. 60, 5294 - 5304 (2005). Cassanello, M., Larachib, F., Kemoun, A., Al-Dahhan, M. H., Dudukovic, M. P., “Inferring liquid chaotic dynamics in bubble columns using CARPT”, Chem. Eng. Sci. Vol. 56, 6125–6134 (2001). Chaumat, H., Billet, A-M., Delmas, H., “Axial and radial investigation of hydrodynamics in a bubble column; Influence of fluids flow rates and sparger type”, Int. J. Chem. Reactor Eng., Vol. 4, No. A25 (2006). Chen, R. C. and Fan, L. S., “Particle image velocimetry for characterizing the flow structures in three-dimensional gas-liquid-solid fluidized beds”, Chem. Eng. Sci. Vol. 47, 3615-3622. (1992). Chen, P., Dudukovic, M. P., Sanyal, J., “Three-Dimensional Simulation of Bubble Column Flows with Bubble Coalescence and Breakup”, AIChE J., Vol. 51, No. 3, 696-712 (2005). Chen, J., Gupta, P., Degaleesan, S., Al-Dahhan, M. H., Dudukovic, M. P., Toseland, B. A., “Gas holdup distributions in large-diameter bubble columns measured by computed tomography. Flow Measurement and instrumentation”, Vol 9, 91-101 (1998). Chen, J., Degaleesan, S., Gupta, P., Al-Dahhan, M.H., Dudulovic, M. P., Toseland, B. A., “Fluid dynamic parameters in bubble columns with internals”, Chem. Eng. Sci., Vol. 54, 2187-2197, (1999). Clift, R., Grace, J. R., Weber, M. E., “Bubbles, Drops, and Particles”, Academic Press, Inc., London, (1978). Deen, N. G., Westerweel, J., Delnoij, E., “Two-Phase PIV in Bubbly Flows: Status and Trends”, Chem. Eng. Tech., Vol. 25, No. 1, 97-101 (2002). Delnoij, E., Westerweel, J., Deen, N. G., Kuipers, J. A. M., Van Swaaij, W. P. M., “Ensemble correlation PIV applied to bubble plumes rising in a bubble column”, Chem. Eng. Sci., Vol. 54, 5159-5171 (1999). Devanathan, N., Moslemian, D., Dudukovic, M. P., “Flow mapping in bubble columns using CARPT”, Chem. Eng. Sci., Vol. 45, 2285-2291 (1990). Dhotre, M. T., Joshi, J. B., “Design of a gas distributor: Three-dimensional CFD simulation of a coupled system consisting of a gas chamber and a bubble column”, Chem. Eng. J., Vol. 125, No. 3, 149-163 (2007). Ekambara, K. , Dhotre, M. T. , Joshi, J. B., “CFD simulations of bubble column reactors:1D, 2Dand 3Dapproach”, Chem. Eng. Sci., Vol. 60, 6733 - 6746 (2005). Ishii, M., Zuber, N., “Drag Coefficient and Relative Velocity in Bubbly, Droplet or Particulate Flows”, AIChE J., Vol. 25, 843-855 (1979). Jakobsen, H., Sannes, B., Grevskott, S., Svendsen, H., “Modeling of vertical bubble-driven flows”, Ind. Eng. Chem. Res., Vol. 36, No. 10, 4052–4074 (1997). Jakobsen, H. A. , Lindborg, H. , Dorao, C. A. , “Modeling of Bubble Column Reactors: Progress and Limitations”, Ind. Eng. Chem. Res., Vol. 44, 5107-5151 (2005). Joshi, J. B., Ranade, V. V., “Computational Fluid Dynamics for Designing Process Equipment: Expectations, Current Status, and Path Forward”, Ind. Eng. Chem. Res., Vol. 42, 1115-1128 (2003). (Commentary) DOI: 10.1021/ie0206608



Joshi, J.B., “Computational flow modelling and design of bubble column reactors”, Chem. Eng. Sci., Vol.56, 5893– 5933 (2001). Kantarci, N., Borak, F., Ulgen, K. O., “Process Biochem”, Vol. 40, 2263., (2005). DOI: 10.1016/j.procbio.2004.10.004 Krishna, R. , Van Baten, J. M., Urseanu, M. I. , “Three-phase Eulerian simulations of bubble column reactors operating in the churn-turbulent regime: a scale up strategy”, Chem. Eng. Sci., Vol.55, 3275-3286 (2000). Krishna, R. , Van Baten, J. M. “Eulerian Simulation Strategy for Scaling up a Bubble Column Slurry Reactor for Fischer-Tropsch Synthesis”, Ind. Eng. Chem. Res., Vol. 43, 4483-4493 (2004). Krishna, R. “A Scale-up Strategy for a Commercial Scale Bubble Column Slurry Reactor for Fischer-Tropsch Synthesis”, Oil & Gas Sci. and Tech., Vol. 55, No. 4, 359-393 (2000). Krishna, R., Van Baten, J.M., Urseanu, M.I., Ellenberger, J., “A scale up strategy for bubble column slurry reactors”, Catalysis Today, Vol. 66, 199–207 (2001). Kulkarni, A.A., Ekambara, K., Joshi, J.B. “On the development of flow pattern in a bubble column reactor: Experiments and CFD”, Chem. Eng. Sci., Vol. 62, No. 4, 1049-1072 (2007). Lain,S., Broder, D., Sommerfeld, M., “Experimental and numerical studies of the hydrodynamics in a bubble column”, Chem. Eng. Sci., Vol. 54, 4913-4920 (1999). Larachi, F., Desvigne, D., Donnat, L., Schweich, D., “Simulating the effects of liquid circulation in bubble columns with internals” Chem. Eng. Sci., Vol. 61, 4195-4206 (2006). Maretto, C., Krishna, R., “Modelling of a bubble column slurry reactor for Fischer-Tropsch synthesis”, Catal. Today, Vol. 52, 279-289 (1999). Matonis, D. , Gidaspow, D., Bahary, M., “CFD Simulation of Flow and Turbulence in a Slurry Bubble Column”, AIChE J., Vol. 48, No. 7, 1413-1429 (2002). Olmos, E., Gentric, C., Vial, C., Wild, G., Midoux, N., “Numerical simulation of multiphase flow in bubble column reactors. Influence of bubble coalescence and break-up”, Chem. Eng. Sci., Vol. 56, 6359–6365 (2001). Pfleger, D., Becker, S., “Modelling and simulation of the dynamic flow behaviour in a bubble column”, Chem. Eng. Sci., Vol. 56, 1737-1747 (2001). Rafique, M., Chen, P., Dudukovic, M., “Computational modelling of gas–liquid flow in bubble columns”. Review in Chemical Engineering, Vol. 20, No.3-4, 225–375 (2004). Ranade, V. V., Tayalia, Y., “Modelling of fluid dynamics and mixing in shallow bubble column reactors: influence of sparger design”, Chem. Eng. Sci., Vol. 56, 1667-1675 (2001). Rigopoulos, S., Jones, A., “A hybrid CFD-reaction engineering framework for multiphase reactor modelling: basic concept and application to bubble column reactors”, Chem. Eng. Sci., Vol. 58, 3077 – 3089 (2003). Santos, C. M., M.Sc. Eng Thesis, State University of Campinas, Brazil, (2005). Sanyal, J. , Vásquez, S. , Roy, S., Dudukovic, M. P., “Numerical simulation of gas-liquid dynamics in cylindrical bubble column reactors”, Chem. Eng. Sci., Vol.54, 5071-5083 (1999).


19

20



Sha, Z., Laari, A., Turunen, I., “Multi-Phase-Multi-Size-Group Model for the Inclusion of Population Balances into the CFD Simulation of Gas-Liquid Bubbly Flows”, Chem. Eng. Technol., 29, No. 5, (2006). DOI: 10.1002/ceat.200500386 Sokolichin, A. , Eigenberger, G. , Lapin, A., “Simulation of Buoyancy Driven Bubbly Flow:Established Simplifications and Open Questions”, AIChE J.,Vol. 50, No. 1, 24-45 (2004). Sokolichin, A., Eigenberger, G., “Applicability of the standard k-ε turbulence model to the dynamic simulation of bubble columns- Part I: Detailed numerical simulations”, Chem. Eng. Sci., Vol. 54, 2273-2284 (1999). Sommerfeld, M., Bourloutski, E., Broder, D., “Euler/Lagrange calculations of bubbly flows with consideration of bubble coalescence”, Can. J. of Chem. Eng., Vol. 81, No. 3-4, 508-518 (2003). Theodoropoulos, C., Sankaranarayanan, K., Sundaresan, S., Kevrekidis, I.G., “Coarse bifurcation studies of bubble flow lattice Boltzmann simulations”, Chem. Eng. Sci., Vol. 59, 2357 – 2362 (2004). Treybal, R. E. “Mass Transfer Operations”, McGraw-Hill Book Co. Mexico, (1980). Van Baten, J.M., Ellenberger, J., Krishna, R., “Scale-up strategy for bubble column slurry reactors using CFD simulations”, Catalysis Today, Vol. 79–80, 259–265 (2003). Van Baten, J. M., Krishna, R., “CFD Modeling of a Bubble Column reactor Carrying out a Consecutive A->B->C Reaction”, Chem. Eng. and Tech. Vol. 27, No.5, 398-406 (2004). Van Baten J. M., Krishna, R., “Eulerian Simulation Strategy for Scaling up a Bubble Column Slurry Reactor for Fischer-Tropsch Synthesis”, Ind. Eng. Chem. Res., Vol. 43, 4483-4493 ( 2004). Van den Hengel, E. I. V., Deen, N. G. , Kuipers, J. A. M., “Application of Coalescence and Breakup Models in a Discrete Bubble Model for Bubble Columns”, Ind. Eng. Chem. Res., vol. 44, 5233-5245 (2005) Wiemann, D., Mewes, D., “Prediction of Backmixing and Mass Transfer in Bubble Columns Using a Multifluid Model”, Ind. Eng. Chem. Res., Vol. 44, 4959-4967 (2005). Wolf-Dieter Deckwer “Bubble Column Reactors”, John Wiley & Sons, New York (1992). Xu, S.K., Qu, Y.H., Chaouki, J., Guy, C., “Characterization of homogeneity of bubble flows in bubble columns using RPT and fibre optics, Int. J. Chem. React. Eng., vol. 3 (2005). Zaruba, A., Kreppera, E. , Prasser, H. M., Vanga, B.N. R., “Experimental study on bubble motion in a rectangular bubble column using high-speed video observations”, Flow Measurement and Instrumentation Vol. 16, 277–287 (2005). Zhang, X., Ahmadi, G. , “Eulerian–Lagrangian simulations of liquid–gas–solid flows in three-phase slurry reactors”. Chem. Eng. Sci., Vol. 60, 5089-5104 (2005). Zheng, Y., Zhang, Q., “Simultaneous measurement ofgas and solid holdups in multiphase systems using ultrasonic technique”, Chem. Eng. Sci. Vol. 59, 3505 – 3514 (2004).


Computational Fluid Dynamics for Designing Process Equipment

Recommend Documents